Re: subrings of ZxZ

In article <518cbdf5-ae9c-4590-a023-eec81e4a9ded@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
Luke Wu <LookSkywalker@xxxxxxxxx> wrote:
What are all the subrings of ZxZ if we require (0,0) and (1,1) to be
in the subring?

the whole ring is a subring

subset of all pairs of form (a,a) is a subring

I believe that's all. Am I right?

No. For example, what about the collection of all pairs (a,b) in which
a-b is a multiple of 5?

This contains (0,0) and (1,1), and (a,a) for all a. If b=a+5k and
d=c+5m, then (a,b) + (c,d) = (a,a+5k) + (c,c+5m) = (a+c,(a+c)+5(k+m)),
so the set is closed under sums. It is also closed under differences;
what about products?

"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes" by Bill Watterson)

Arturo Magidin


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